Advanced Search

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho$-integral, introduced by Jarník and Kurzweil. Let $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ be the space of all strongly $\rho$-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel ·\parallel$. We show that each element in the dual space of $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ can be represented as a strong $\rho$-integral. Consequently, we prove that $fg$ is strongly $\rho$-integrable on $E$ for each strongly $\rho$-integrable function $f$ if and only if $g$ is almost everywhere...

Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.

In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is $F_{\sigma \delta }$ regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.

It is shown that if $g$ is of bounded variation in the sense of Hardy-Krause on $\prod _{i=1}^m[a_i,b_i]$, then $g{\chi }_{{}_{\prod _{i=1}^m(a_i,b_i)}}$ is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.

Several new integrability theorems are proved for multiple cosine or sine series.

Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.

We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.

We use an elementary method to prove that each $BV$ function is a multiplier for the $C$-integral.

Some full characterizations of the strong McShane integral are obtained.

It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$-dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f{[0,1]}^2⟶ℝ$ and a continuous function $F{[0,1]}^2⟶ℝ$ such that $$\left(\P \right){\int }_0^x\left\{\left(\P \right){\int }_0^{y}f(u,v)\mathrm{d}v\right\}\mathrm{d}u=\left(\P \right){\int }_0^y\left\{\left(\P \right){\int }_0^{x}f(u,v)\mathrm{d}u\right\}\mathrm{d}v=F(x,y)$$ for all $(x,y)\in {[0,1]}^2$.